Sistemas y Señales Biomédicos

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2025-09-29

Sistemas y Señales Biomedicos - SYSB

Digital Filter – Introduction

  • It is a mathematical algorithm or system that processes digital signals.
  • They enhance, suppress, or modify specific frequency components.
  • These filters are essential for removing noise, extracting relevant information, and improving signal quality.

What is a system?

A system is a rule that maps an input signal to an output signal. In continuous time and discrete time we depict and denote: \[ x(t)\ \longrightarrow\ y(t),\qquad x[n]\ \longrightarrow\ y[n]. \] These are the standard input–output representations used throughout the text.

Input–output operator notation

We often write the system as an operator ( {} ): \[ y(t)=\mathcal{T}\{x(t)\},\qquad y[n]=\mathcal{T}\{x[n]\}. \] Block diagrams are used to represent systems and interconnections (series/cascade and parallel).

Interconnections (examples)

  • Series (cascade): output of System 1 feeds System 2.
  • Parallel: both systems process the same input; outputs are summed. These patterns are foundational for building complex systems from simpler ones.

Property: Memoryless vs. with memory

  • Memoryless: the output at an instant depends only on the input at that same instant.
  • With memory: the output depends on past/future values (e.g., accumulators, averagers). Example: the summer/accumulator (running sum) and its inverse (first difference) illustrate systems with memory: \[ y[n]=\sum_{k=-\infty}^{n} x[k],\qquad \text{inverse: } y[n]=x[n]-x[n-1]. \] Both are causal (see next slide)

Property: Causality (general and LTI)

Causal: output depends only on present/past input values. For LTI systems, causality is characterized by the impulse response: \[ \text{Discrete time: } h[n]=0\ \text{for } n<0;\qquad \text{Continuous time: } h(t)=0\ \text{for } t<0. \] Under these conditions, the convolution reduces to depend only on past/present input.

Property: Stability (BIBO)

Bounded-Input Bounded-Output (BIBO) stability: bounded input implies bounded output. For LTI systems: \[ \sum_{k=-\infty}^{\infty} |h[k]| < \infty \quad \Longleftrightarrow \quad \text{discrete-time LTI is stable}, \] \[ \int_{-\infty}^{\infty} |h(t)|\,dt < \infty \quad \Longleftrightarrow \quad \text{continuous-time LTI is stable}. \] These are necessary and sufficient conditions.

Property: Linearity (superposition)

A system is linear if it satisfies additivity and homogeneity: for any signals (x_1,x_2) and scalar (c), \[ \mathcal{T}\{x_1+x_2\}=\mathcal{T}\{x_1\}+\mathcal{T}\{x_2\},\qquad \mathcal{T}\{c\,x\}=c\,\mathcal{T}\{x\}. \] (These two conditions together are equivalent to linearity.)

Property: Time invariance (definition)

A system is time-invariant if a shift in the input produces an identical shift in the output: \[ \mathcal{T}\{x(t-t_0)\}=y(t-t_0),\ \text{whenever}\ y(t)=\mathcal{T}\{x(t)\}. \] Analogously in discrete time with shifts by integer indices. (Definition used throughout the text in system properties and LTI analysis.)

Property: Invertibility

A system is invertible if distinct inputs produce distinct outputs; equivalently, there exists an inverse system that recovers the input from the output. Example pair (discrete time): the accumulator and the first-difference operator are inverses: \[ y[n]=\sum_{k=-\infty}^{n} x[k] \quad \Longleftrightarrow \quad x[n]=y[n]-y[n-1]. \]

Additional note: Interconnections and LTI analysis

For LTI systems, interconnections admit simple algebraic descriptions via transforms: e.g., in the Laplace domain, series and parallel lead to product and sum of system functions, respectively: \[ H_{\text{series}}(s)=H_1(s)H_2(s),\qquad H_{\text{parallel}}(s)=H_1(s)+H_2(s). \]

Digital Filter – Introduction

Importante

The digital filter separates the noise and the information of a discrete signal.

Digital Filter – Introduction

Digital Filter – Introduction

Suppose a discrete time system \[ y[n] = \sum_{k=1}^{K} a_k y[n - k] + \sum_{m=0}^{M} b_m x[n - m]\]

  • K y M are the order of the filter.

  • We must know the initial condition.

Examples of digital filters

Gain

\[y[n] = G x[n]\]

Delay of \(n_0\) samples

\[y[n] = x[n - n_0]\]

Two points moving average

\[y[n] = \frac{1}{2} (x[n] + x[n - 1])\]

Euler approximation of the derivative

\[y[n] = \frac{x[n] - x[n - 1]}{T_s}\]

Averaging over N consecutive epochs of duration L

\[y[n] = \frac{1}{N} \sum_{k=0}^{N-1} x[n - kL]\]

Trapezoidal integration formula

\[y[n] = y[n - 1] + \frac{T_s}{2} (x[n] + x[n - 1])\]

Digital “leaky integrator” (First-order lowpass filter)

\[y[n] = a y[n - 1] + x[n], \quad 0 < a < 1\]

Digital resonator (Second-order system)

\[y[n] = a_1 y[n - 1] + a_2 y[n - 2] + b x[n], \quad a_1^2 + 4a_2 < 0\]

The impulse response

  • The impulse response, denoted as \(ℎ[n]\), is the output of a digital filter when the input is a unit impulse function \(\delta[n]\)
  • The impulse response fully describes the system. Given \(h[n]\), we can determine the output for any input using convolution.
  • Different types of filters (low-pass, high-pass, band-pass, etc.) have characteristic impulse responses.

Conditions

For a system’s response to be fully described by its impulse response, the system must satisfy the following key conditions.

Linearity

If the system responds to \(x_1[n]\) with \(y_1[n]\) and to \(x_2[n]\) with \(y_2[n]\), then:

\[y[n] = y_1[n] + y_2[n]\]

Homogeneity

If the input is scaled by a constant \(c\), the output is also scaled:

\[\text{If } x[n] \rightarrow y[n], \text{ then } cx[n] \rightarrow cy[n]\]

Time Invariance

A system must be time-invariant, meaning a time shift in the input causes the same shift in the output:

\[\text{If } x[n] \rightarrow y[n], \text{ then } x[n - n_0] \rightarrow y[n - n_0]\]

Causality

A causal system is one where the output at time \(n\) depends only on present and past inputs:

\[h[n] = 0 \quad \forall n < 0\]

Stability

If the impulse response does not satisfy this condition, the system may produce unbounded outputs.

\[\sum_{n=-\infty}^{\infty} |h[n]| < \infty\]

Convolution Representation

If all condition met then \[y[n] = x[n] * h[n] = \sum_{m=-\infty}^{\infty} x[m] h[n - m]\]

Convolution